Problem 82 - Algebra and Trigonometry

Question

Find the indicated roots, and graph the roots in the complex plane. The cube roots of \(4 \sqrt{3}+4 i\)

Step-by-Step Solution

Answer

The cube roots are found by converting to polar form, dividing the argument by 3, and using roots of unity. Plot them at equal angles on a circle of radius 2 in the complex plane.

Step 1: Convert to Polar Form

First, we need to convert the complex number from rectangular form to polar form. Given the complex number \( z = 4\sqrt{3} + 4i \), we need to find the modulus \( r \) and the argument \( \theta \). The modulus is found using \( r = \sqrt{(4\sqrt{3})^2 + 4^2} = \sqrt{48 + 16} = \sqrt{64} = 8 \). The argument \( \theta \) is given by \( \theta = \tan^{-1}\left(\frac{4}{4\sqrt{3}}\right) = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \). Thus, the polar form is \( 8(\cos(\frac{\pi}{6}) + i\sin(\frac{\pi}{6})) \).

Step 2: Find Principal Cube Root

To find the cube roots, we calculate \( \sqrt[3]{8} = 2 \) for the modulus, and divide the argument by 3. This gives us the principal cube root \( 2(\cos(\frac{\pi}{18}) + i\sin(\frac{\pi}{18})) \).

Step 3: Use the Roots of Unity

Each cube root can be found by multiplying by the cube roots of unity, which are \( 1, \omega = e^{2\pi i / 3}, \omega^2 = e^{4\pi i / 3} \). Therefore, the cube roots are \( z_1 = 2(\cos(\frac{\pi}{18}) + i\sin(\frac{\pi}{18})) \), \( z_2 = 2(\cos(\frac{\pi}{18} + \frac{2\pi}{3}) + i\sin(\frac{\pi}{18} + \frac{2\pi}{3})) \), and \( z_3 = 2(\cos(\frac{\pi}{18} + \frac{4\pi}{3}) + i\sin(\frac{\pi}{18} + \frac{4\pi}{3})) \).

Step 4: Convert Back to Rectangular Form

Convert each of the roots from polar form back to rectangular form for plotting. Use the expressions: \( z_1 = 2\cos(\frac{\pi}{18}) + i \times 2\sin(\frac{\pi}{18}) \), \( z_2 = 2\cos(\frac{13\pi}{18}) + i \times 2\sin(\frac{13\pi}{18}) \), \( z_3 = 2\cos(\frac{25\pi}{18}) + i \times 2\sin(\frac{25\pi}{18}) \). Calculate these values to get approximate rectangular coordinates.

Step 5: Plot in the Complex Plane

Plot the cube roots on the complex plane using their rectangular coordinates. The angle increments and symmetric placement around the origin allow easy graphical representation. Ensure the plot shows equal spacing around a circle of radius 2, centered at the origin.

Key Concepts

Polar Form

The polar form of a complex number provides an alternative to the rectangular form, which is typically expressed as \( a + bi \). In polar form, any complex number can be represented using a combination of its modulus (length) and argument (angle). This is particularly useful in multiplying and dividing complex numbers.
For a given complex number \( z = x + yi \):

• The modulus \( r \) is given by \( r = \sqrt{x^2 + y^2} \).
• The argument \( \theta \) is the angle formed with the positive x-axis, calculated using \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \).

Once \( r \) and \( \theta \) are determined, the polar form becomes \( z = r(\cos \theta + i \sin \theta) \), sometimes abbreviated using Euler's formula as \( re^{i\theta} \).
This transformation simplifies operations like exponentiation and root extraction, making it easier to work with complex numbers in advanced mathematics.

Cube Roots

Finding the cube roots of a complex number involves determining all the possible values that, when cubed, return the original complex number. In polar form, calculating these roots is straightforward and intuitive.

First, derive the principal cube root. This is accomplished by taking the cube root of the modulus \( r \) and dividing the argument \( \theta \) by 3. For a complex number in polar form \( z = r(\cos \theta + i \sin \theta) \):

• New modulus: \( \sqrt[3]{r} \).
• New argument: \( \frac{\theta}{3} \).

Next, employ the roots of unity to locate all cube roots. Because each root is separated by equal angles, the roots of the number are expressed as: \( z_k = \sqrt[3]{r}(\cos(\frac{\theta}{3} + \frac{2k\pi}{3}) + i\sin(\frac{\theta}{3} + \frac{2k\pi}{3})) \) for \( k = 0, 1, 2 \).
This method reveals the symmetry of root placement, making it visually clear on the complex plane.

Roots of Unity

Roots of unity are foundational in understanding the cyclical nature of complex numbers, particularly when finding roots like the cube roots. The \( n \)-th roots of unity are special complex numbers that satisfy the equation \( z^n = 1 \).

In polar form, these roots are spaced evenly around the unit circle as follows:

• The \( n^{\text{th}} \) root of unity for each \( k \) is \( e^{2\pi ik/n} \), giving rise to the tetrahedral symmetry.

This matters when determining the full set of solutions for equations involving powers of complex numbers. To find, say, cube roots of a number \( z \), multiply the principal root by these roots of unity:

• \( \omega = e^{2\pi i/3} \)
• \( \omega^2 = e^{4\pi i/3} \)

These operations neatly place the roots evenly around a circle in the complex plane, assisting in finding every root clearly and systematically.

Complex Plane

The complex plane is a visual tool to represent and work with complex numbers, akin to a modified Cartesian plane where the x-axis represents real numbers and the y-axis represents imaginary numbers. Each complex number \( a + bi \) is a point \((a, b)\) on this plane.

When discussing cube roots:

• The roots typically take on a distinct geometric shape, often spaced symmetrically around a circle.
• These roots can be easily plotted by converting from polar back to rectangular coordinates.

This visual layout helps display relationships between roots clearly. It highlights symmetry and equidistance unique to complex number operations. Plotting them can make understanding complex analysis concepts more intuitive by showing how different roots correspond to different points on the plane, often around a centered circle with a specific radius.